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Support No mathematical proposition can be proven true by observation. ██ ███████ ████ ██ ██ ██████████ ██ ████ ███ ████████████ ███████████ ██ ██ █████
Summary
The author concludes that mathematical propositions cannot be known to be true. This is based on the fact that mathematical propositions cannot be proven true by observation.
Missing Connection
There’s a difference between being impossible to prove true by observation and being impossible to know is true. What if mathematical propositions are impossible to prove true by observation, but are possible to know through some other method besides observation, such as through logic? This is the flaw in the argument.
To make the argument valid, we want to know that if something is impossible to prove true by observation, then it is impossible to know is true. In other words, we want to know that in order to know that something is true, we must be able to prove it true through observation.
24.
The conclusion follows logically if █████ ███ ██ ███ █████████ ██ ████████
a
Only propositions that ███ ██ ██████ ████ ███ ██ █████ ██ ██ █████
(A) doesn’t make the argument valid, because we don’t know that mathematical propositions can never be proven true. We do know that they can’t be proven true through observation; but they might be provable through some other method besides observation.
b
Observation alone cannot ██ ████ ██ █████ ███ █████ ██ ███ ████████████
(B) leaves open the possibility that we might be able to know the truth of math propositions through other methods besides observation.
c
If a proposition ███ ██ ██████ ████ ██ ████████████ ████ ██ ███ ██ █████ ██ ██ █████
This is the reverse of what we’re looking for. We want to know that in order for something to be known true, it must be provable by observation. But (C) tells us that provability by observation is sufficient to know something is true. That doesn’t establish that if something is impossible to prove through observation, we can’t know it’s true.
d
Knowing a proposition ██ ██ ████ ██ ██████████ ████ ██ ██ ██████ ██ ██████ ████ ██ ████████████
This is the reverse of what we’re looking for. (D) tells us that if we can’t know something to be true, then it can’t be proven true by observation. But (D) leaves open the possibility that some things that can’t be proven true by observation might still be knowable as true through some other method.
e
Knowing a proposition ██ ██ ████ ████████ ███████ ██ ████ ██ ████████████
(E) tells us that provability through observation is necessary in order to know a proposition to be true. This establishes that if we can’t prove something true through observation — as is the case with math propositions — then we can’t know it to be true.